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212.] Again, let us take the integral (10) to be the subject of transformation, and suppose the n equations connecting 1, 2, X, 1, 2,... En to be implicit, and to be of the forms

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f1 = 0, f2 = 0,
f2

f1 = 0;

(21)

also let the several partial derived functions of these be denoted by letters according to the following scheme; viz.,

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a1dx1+b1dx 2 + ... + r„ dxn+a„ d§1 + ... + Pn d§n = 0.

(22)

Now when 1 varies, dx2 = dx3 = ... = dx1 = 0; so that in calculation the quantity which is to replace da1, we have

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now substituting this value for da, in the given element of integration, and eliminating 1, 2, 3, ... §. from that element by means of (21), it becomes a function of §1, X2, X3, Xn; and consequently 1, 3, ... x, are all constant, when 2 varies; so that in calculating the quantity which is to replace dx2,

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and therefore from (22),

a1 dx1 + ẞ1 d2+ ... + P1 dƐn = − b1 dx2,
ρι αξη

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(25)

(26)

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Equation (19) is evidently only a particular case of (28); that =... = r n = 1, and all the other

viz., in which a1 = b2 = C3= partial derived functions vanish.

Hence we have the following theorem ;

In transforming the multiple integral

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into its equivalent in terms of 1, 2, ... En, where these variables are related by the n equations f1 = 0, ƒ1⁄2 = 0, ƒ1⁄2 = 0, . . . ƒ„ = 0 ;

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...ƒn=0;

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The sign of the result being ambiguous, because it depends on the order of integration in the transformed integral, and that order is obviously arbitrary*.

213.] The following are examples illustrative of the preceding theorem.

A full discussion of the theory of the transformation of multiple integrals will be found in a paper on the subject by M. Catalan, in the Mémoires couronnés par l'Académie de Bruxelles, Tome XIV, 1839, 1840.

Ex. 1. Take the double integralƒ ƒr (x, y) dy da; and (1) let

the equations of transformation be

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These cases are obviously those of transformation from a rectangular system of axes, to another rectangular system and to a polar system respectively.

(3) More generally, if the equations of transformation are given in the form,

x = ƒ1(§,n), y = f2 (§, n); then

dx = (df)
(d) d€ + (d) dn,

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Ex. 2. Let the integral be the triple integral

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(33)

(34)

and (1) let the equations of transformation be those from a system of rectangular axes to another system of rectangular axes; that is, let

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(2) Let

x = r sin 0 cos 4,

(36)

y = r sin 0 sin o, z = r cos 0;

da sin cos o dr+r cos 0 cos & de-r sin 0 sin o do,

dy = sine sin & dr+r cos 0 sin & de+r sin 0 cos & do,
dz = cos 0 dr

.*.

-r sin o do;

▲ = ] ; 41 = r2 sin 0;

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(37)

(38)

These latter equations are those of transformation from a rectangular to a polar system in geometry of three dimensions.

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which equation we shall find it convenient to use instead of the last of the group (39); differentiating these with respect to X1, X2, Xn, we have

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Differentiating (40) and (39) with respect to r, 01.01

(41)

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whence 4, (-)"-12" (sin1)"-1 (sine)"-2...sine-1; so that

41

A

= ± TM”−1
-1 (sin 0,)"-2 (sin 0)"-3... sin 0-2 dr do, do do„–1 ;

and consequently,

PRICE, VOL. II.

PP

dx dx2... dxn

± 0,

= "-1(sin1)"-2 (sin 0)"-3...sin 0-2 dr do, do dom-1 (42)

If n =

3, we have

dx dx2 dx3 = r2 sin 01 dr dė1 de2 ;

1

which is the transformation given in the latter part of the preceding example.

Ex. 4. Let the equations of transformation be

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dx3 = §2(1—§3) d§1 + §1 (1 − §3) d§2 — §1 §2 d§3,

(43)

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n-1

.. dx dx2... dxn dx1 = ± §1”−1 §2"-2... En-1 d§1 d§2... dƐn. (44)

Hence we have the following transformation, when n = 2;

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Ex. 5. To transform into its equivalent dx dy dz, when x= lr, y = mr, z = nr; l, m, and n being subject to the condition

12 + m2 + n2 = 1.

By a process similar to those above, we have the following

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214.] If the original integral is definite, the transformed one will also be definite; for the latter is to be equivalent to the former in all respects, and consequently the values of the variables in both integrals must extend over the same district of

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